Base in Naples or Sorrento?

Depends on what you wish to get from the experience. Naples was a great base with walking distance to the ferry terminal to explore the island of Capri and Ischia. The apartments, if you choose Airbnb’s are very plentiful as well for a reasonable cost. The train is in easy reach as well. Other than this...Naples is hustle and bustle! A little fast pace, a little rough and a little dirty for me. I cannot account for the latter as I did not go to Sorrento.

It's easy to read about Sorrento, but here's some (in my view, positive) Naples material:

https://community.ricksteves.com/travel-forum/trip-reports/surviving-naples

If you prefer a bustling large city, Naples. If you prefer a smaller town, Sorrento.
Do you prefer NYC or Newport, RI? Toronto or Quebec City?

They're very, very different, so it depends on what you want. Naples is busier and less tourist-driven, and some people find it noisy and dirty (whereas others find it energetic and fun) - I don't know which you'd be. I loved Naples when I was there for 5 days/6 nights, but, even then I knew it was a nice place to visit but I wouldn't want to live there. Still, I didn't like the feeling of Sorrento at all -- I took the ferry there from Naples and wandered around for a while, and thought it was too touristy. By that, I don't mean there were too many tourists, but that (at least) the main part of the city seemed to be developed and exists for the tourist market. It almost seemed like a disneyland. On the other hand, I know people who've gone for vacation there and loved it........maybe it's one of those places that gets better with time i.e. many people don't like it when they first arrive, but grow to like it as they're there for a while. (I've heard that about Naples and Venice, too.)

Both are good destinations. We chose a week in Sorrento. It was a good base to visit Naples. Herculaneum Pompeii Amalfi coast and on a long day trip Paestum. We had a day in Naples and were intrigued by its grittiness and liveliness but were happy to be in Sorrento.
Maybe these images will be helpful.
Sorrento http://www.flickr.com/photos/stanbr54/sets/72157634654031558/

Naples https://www.flickr.com/photos/stanbr54/albums/72157661599615186

We loved staying in Sorrento. Try to get a hotel that overlooks the Bay of Naples. The sunsets are gorgeous and the Centro Storico area is fun to walk around. Sorrento is a great base to take a ferry to Capri or the train to Pompeii. We split our time between Sorrento and Positano and also took a ferry to Amalfi and then a bus to Ravello. We stayed overnight in Naples before our flight home. The traffic was heavy in Naples and there were many panhandlers there, so we did not feel comfortable. The Archaeological Museum in Naples is well worth it as a side trip.

or Salerno! I have spent about 5 days in each. All have pluses and maybe a minus or two.

Salerno's pluses: Fast train connection to Naples/Rome. Train to Pompeii (though the "rear" entrance). Train or bus to Paestum (Greek temples). A well-preserved centro storico, a large sandy beach, a seaside promenade, very good restaurants. It's cheaper (food and lodging) than Sorrento, though Naples is even less. Buses and ferries to the Amalfi Coast start here.

Sorrento! That is the place.

You sound keen to stay in one place, and if that’s your plan then Sorrento is the easy choice. Naples is a little too far to explore the Amalfi Coast effectively. Naples is certainly worth some time to see though, I pretty much loved it and would like to go back. There isn’t much to Sorrento itself, but it’s transport options can’t be beat. Having options is gold in this area. Sorrento is a good respite to intense sightseeing.

I would suggest though, that with seven days you could split your stay between Naples/Sorrento and an Amalfi Coast town. There’s a REAL benefit to spending a few nights on the coast itself. The coast gets packed with day trippers from Sorrento, Salerno, Naples and beyond. If you stay on the coast, you can avoid much of that and it adds to the enjoyment of the place. If you’re day tripping you can spend a lot of your time commuting. Getting around there is a time eater. The less of it you can do the better.